Theorem sseqtrrd 3267

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
sseqtrrd.1	|- ( ph -> A C_ B )
sseqtrrd.2	|- ( ph -> C = B )

Assertion

Ref	Expression
sseqtrrd	|- ( ph -> A C_ C )

Proof of Theorem sseqtrrd

Step	Hyp	Ref	Expression
1		sseqtrrd.1	. 2 |- ( ph -> A C_ B )
2		sseqtrrd.2	. . 3 |- ( ph -> C = B )
3	2	eqcomd 2237	. 2 |- ( ph -> B = C )
4	1, 3	sseqtrd 3266	1 |- ( ph -> A C_ C )

Syntax hints:   -> wi 4   = wceq 1398   C_ wss 3201

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-in 3207  df-ss 3214

This theorem is referenced by:  sseqtrrid  3279  fnfvima  5899  tfrlemiubacc  6539  tfr1onlemubacc  6555  tfrcllemubacc  6568  rdgivallem  6590  nnnninf  7368  nninfwlpoimlemg  7417  ccatass  11234  swrdval2  11281  dfphi2  12855  ctinf  13114  imasaddfnlemg  13460  imasaddvallemg  13461  subsubm  13629  subsubg  13847  subsubrng  14292  subsubrg  14323  lidlss  14555  toponss  14820  ssntr  14916  iscnp3  14997  cnprcl2k  15000  tgcn  15002  tgcnp  15003  ssidcn  15004  cncnp  15024  txcnp  15065  imasnopn  15093  hmeontr  15107  blssec  15232  blssopn  15279  xmettx  15304  metcnp  15306  plyaddlem1  15541  plymullem1  15542  plycoeid3  15551  nnsf  16714  nninfsellemsuc  16721